Time evolution of impurity models and their universality for quantum computation

• URL: http://arxiv.org/abs/2604.08466v1
• Date: Thu, 09 Apr 2026 17:01:59 GMT
• Title: Time evolution of impurity models and their universality for quantum computation
• Abstract summary: Impurity Hamiltonians are systems of $N$ fermionic modes where $O(1)$ of them interact among themselves via quartic (or higher order) fermion terms.<n>We prove that the time evolution of generic time-independent impurity Hamiltonians on $O(N)$ qubits is universal on $N$ qubits if the input state is a product state of fermions in any single particle basis.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Impurity Hamiltonians are systems of $N$ fermionic modes where $O(1)$ of them interact among themselves via quartic (or higher order) fermion terms, while coupling quadratically with $O(N)$ bath modes. Without the quartic interactions, these systems are classically simulable with $O(N^3)$ resources. It was proved that the time-dependent evolution of these systems can perform universal quantum computation. The question of whether or not this remains true for time-independent evolution remains open. Here, we prove that the time evolution of generic time-independent impurity Hamiltonians on $O(N)$ qubits is universal on $N$ qubits if the input state is a product state of fermions in any single particle basis. In our proof we find that for a computation of depth $S$, the size of the impurity scales as $O(S\log S)$.

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